MTH515A - Inference - II

Group families, the principle of equivariance, location family, scale family, location scale family. Minimum risk equivariance estimators, risk functions, admissibility, prior distribution, posterior distribution, geometric interpretation for finite parameter space, Bayes estimators, limit of Bayes estimators, minimax estimators and their relations. Review of convergence in probability and convergence in distributions. Consistency results of the mle's, and the mme's. Asymptotic relative efficiency. Consistent and Asymptotic Normal (CAN) estimators, Invariance of CAN estimators under different transformations. CAN estimators obtained by moments and MLE methods in one parameter xponential family and multi parameter exponential family. Sequential Probability Ratio Tests and its applications in different practical problems. Invariant test and unbiased tests, Likelihood ratio test and its asymptotic distributions, Wald test, Rao's score test, Pearson c2 test for goodness of fit. Large sample tests and confidence intervals based on CAN estimators. Consistency of large sample tests and asymptotic powers of large sample tests.